In [2] an alternative skolemization method called eskolemization was introduced that is sound and complete for existence logic with respect to existential quantifiers. Existence logic is a conservative extension of intuitionistic logic by an existence predicate. Therefore eskolemization provides a skolemization method for intuitionistic logic as well. All proofs in [2] were semantical. In this paper a proof-theoretic proof of the completeness of eskolemization with respect to existential quantifiers is presented.